Nil-generated algebras and group algebras whose units satify a Laurent polynomial identity

Let A be an algebra whose group of units U(A) satisfies a Laurent polynomial identity (LPI). We establish conditions on these polynomials in such a way that nil-generated algebras and group algebras with torsion groups over infinite fields in characteristic p > 0 have nonmatrix identities. We also determine, in the determine in the context of group algebras with arbitrary LPI for the group of units, the existence of polynomial identities.